Note that a regular p-gon can be dissected into p isosceles triangles with vertex angle 2p/p, so base angles (p/2)(1-2/p) and two of these fit together at each corner, so the angle at each corner is p(1-2/p). Hence a regular polyhedron with symbol {p,q} has total face angle measure at each corner of  p q(1-2/p). But in order to form a closed convex body, the measures of these q face angles at a corner must make a total < 2p.

 p is at least 3,  giving face angles at least p/3, and if p=3 the possibilities for q satisfying this restraint are 3, 4 or 5. For p=4 we get q=3 as the only possibility and for p=5, the only possibility is q=3.  Thus the the only possibilities are {p,q} = {3,3}, {4,3}, {3,4}, {5,3} or {3,5}. The symbol completely specifies the regular polyhedron, and we have one for each symbol, so there are exactly 5 regular polyhedra. 

Here are some pictures of them. Apart from an orthogonal projection, you will see a net for constructing the skeleton, the symbol and the Schlegel diagram which is a projection from a point close to one face, and from which it is easy to study the relationship between corners, edges and faces.

Dual Polyhedra

 To construct a regular polyhedron dual to a {p,q}, put a point in the centre of each face and join a pair of points by a line segment whenever the corresponding faces meet in an edge. This new figure clearly has the same number of corners as the original had faces, the same number of edges as the original, and the same number of faces as the original had corners. So it is a {q,p}. In this way you have embedded an octahedron in a cube and a cube in an octhedron, a dodecahedron in an icosohedron and vice versa, and a tetrahedron in a tetrahedron. Furthermore, every symmetry of the original is a symmetry of the new polyhedron, or in other words, the symmetry group O₃ of the cube is the symmetry group of the octahedron, and the same for the dodecahedron and the icosohedron. We have cut down our work by almost half!

Symmetry groups of the Regular Polyhedra

There can be no more rigid motions of the tetrahedron: there are four ways to choose which face is at the bottom, and when chosen, it can be in any of 3 positions. Thus there are 12 possibilities and we have found 12 different rigid motions.

 In order to see whether this group is isomorphic to a known group, label the corners {1,2,3,4} and consider the action of the rigid motions on this set, i.e look at the corresponding homomorphism into S₄.

1. Type 0 -> (1)
2. Type 1 -> (123) and the other 3-cycles
3. Type 2 -> (12)(34) the other products of two disjoint transpositions.

 To see geometrically what is happening to the tetrahedron, notice that (123) is caused by a rotation of (2p)/3 about the axis running from corner 4 to the centroid of the opposite face, and (132) by a rotation of (4p)/3 about the same axis.

 Then the rigid motion (12)(34) is caused by a rotation of magnitude p about the axis running from the midpoint of the edge 12 to the midpoint of the edge 34. For future reference, note that if s = (12)(34) and a = (123), then s^-1.a.s = (143), which is rotation by (2p)/3 about the axis running from corner 2 to the centroid of the opposite face. In other words, conjugating by s has the effect of transforming a rotation about some axis to rotation by the same amount about the image of this axis under s.

 To complete the analysis, reflect the tetrahedron in the plane passing through two corners and the mid-point of the opposite side. This changes the orientation of edges of each face, so gives the other 12 permutations that define S₄.

 We will consider the symmetry group of the dodecahedron and the icosohedron in Section 8. Other polyhedra which are not regular, but have a high amount of symmetry can be constructed by truncating the corners of the regular ones. For some examples, click here and here.
 

Author: Phill Schultz, schultz@maths.uwa.edu.au